Linear Motility Maps in Nonlinear Viscous Fluids

Robotic locomotion can become efficient when mechanisms exploit passive dynamics, compliance, and resonance rather than track prescribed trajectories. This paper formulates natural locomotion as an exchange principle for systems whose motion is mediated by environmental constraints or interactions. A motion is natural when an internal oscillator returns periodically, the body pose drifts, and the mean Propulsion--Oscillator Exchange power (POE power) vanishes over one cycle. The selected family is a Natural Locomotion Manifold (NLM). We develop the conservative realization of this principle for continuous ideal environmental constraints: the constraints do no external work, total mechanical energy is conserved, and zero mean POE power is an internal exchange with the environment-mediated propulsive channel, not external energy input. The method is a closed/open construction. The propulsive channel is first closed to reveal an effective internal oscillator, organized by scalar action-angle structure in one effective degree of freedom or by nonlinear modal sectors in several degrees of freedom. The channel is then reopened, pose is reconstructed, and accepted cycles must preserve internal recurrence and zero mean POE power. We demonstrate the principle on two ideal nonholonomic no-slip systems: a Chaplygin-sleigh / pendulum-driven car and a three-body extension. In the scalar case, POE closure is equivalent to the missing internal return condition, giving a theorem-backed computation of the NLM family. In the multi-degree case, POE closure remains necessary but must be completed by modal identity, internal return, dynamics consistency, same fixed passive architecture, and nonzero displacement. Natural locomotion becomes a design question: which passive architectures support no, one, or several certified NLM families?

Animals and robots encounter mud at the water-land interface. Like sand, mud can stay solid or flow like a fluid. Unlike sand, the yield strength of mud at which solid-fluid transitions occur depends on not only the amount of solid relative to fluid (water in mud, air in dry sand), but also how much coarse grains and fine clay are within the solid. Despite understanding of locomotion on/within dry sand dominated by coarse grains with repulsive normal forces and friction, little is known for mud dominated by fine clay with strong cohesion. Here, we developed methods to prepare uniform mud of controlled, variable yield strength and characterize and track its drift from water evaporation. Compared to other flowable substrates, mud strength measured by upward force during penetration is weaker and can vary more, and mud sticks more during extraction to pull downward, making it more challenging for locomotion.

Efficient flapping propulsion hinges on operating within a narrow Strouhal number window, a principle nature has converged upon for maximum thrust-to-power ratio. We translate this bioinspired empirical rule into real-time control, demonstrating it on an autonomous underwater vehicle driven by four soft fins. The proposed Strouhal-aware Model Predictive Control (MPC) enhances a quasi-steady hydrodynamic model with an explicit penalty for Strouhal deviation, solving the resulting nonconvex problem via a two-stage sampling and gradient optimization that runs onboard at 25 Hz. Pool and field trials show that the controller keeps each fin within the optimal Strouhal corridor (0.25-0.35) while precisely tracking commanded forces. This results in a mean reduction in mechanical power of 8.8% to 32% throughout the cruising range of 0.1 to 0.3 m/s. The proposed method also allows for a velocity of 0.4 m/s, which is unattainable for a baseline of the conventional inverse model. The results confirm that embedding first-principle flow physics into an MPC objective yields tangible endurance gains without sacrificing agility, offering a generic pathway to energy-aware locomotion in next-generation multifin robots.